Moment expansion for mapping of the confined diffusion

The mapping of diffusion in a 2D channel with varying cross section $A\left(x\right)$ onto the longitudinal coordinate $x$ is revisited. We present an algorithm based on construction of a specific hierarchy of equations for the transverse moments of the 2D density. Elimination of all the moments but the zeroth one, the 1D density $p(x,t)$, results in the mapped equation. Our calculation validates the earlier mapping procedure [P. Kalinay and J. K. Percus, Phys. Rev. E 74, 041203 (2006); P. Kalinay and J. K. Percus, Phys. Rev. E 78, 021103 (2008)], presuming existence of the backward mapping operator, and it naturally arrives at the extended Fick-Jacobs equation [D. Reguera and J. M. Rubì, Phys. Rev. E 64, 061106 (2001)] in the stationary flow, without any phenomenological conjectures.

Figure 1

The channel bounded by $0<y<A\left(x\right)$. The flux density $j(x,y,t)$ at the boundaries is parallel to them, giving the Neumann BC, Eq. (1.2). The aim is to find an evolution equation for the 1D density $p(x,t)$ defined by Eq. (1.3).

Figure 2

Plots of the effective diffusion coefficient $D\left(x\right)$ depending on the slope of the boundary $A^{\prime }\left(x\right)$ according to the approximate formula Eq. (3.7) (dashed line), the formula of Reguera and Rubì, $D\left(x\right)={(1+A^{\prime 2})}^{-1/3}$ (dotted line) and the exact coefficient for the linear cone, $D\left(x\right)=\arctan \left(A^{\prime }\right)/A^{\prime }$ (full line).